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Forming the equation of a line, given the coordinates of a point on it, and its slope

Given the coordinates of a point (x1, y1) that lies on a particular line, and the slope of the line 'm', its equation can be obtained by the following
(y - y1) = m(x - x1)
This is known as the point-slope formula, or the point-slope form of the equation of a line. On substituting the values of x1, y1 and 'm' in the above equation, the equation of a line, containing the point (x1, y1) and having a slope of 'm', is derived.


For example, give a point (1, 10) and the slope m = 2 for a line, its equation is derived as follows:
(y - y1) = m(x - x1)
(y - 10) = 2(x - 1)
The values of x1 , y1 are obtained from the point (1, 10), where x1 = 1 and y1 = 10, while 'm' is the slope of the line, given 2.

Simplifying the above equation, and writing it in slope-intercept form,
(y - 10) = 2(x - 1)
y - 10 = 2x - 2
y = 2x - 2 + 10
y = 2x + 8,
which is the equation of the line containing the point (1, 10) and having a slope of 2.

Writing the equation in standard form,
(y - 10) = 2(x - 1)
y - 10 = 2x - 2
-2x + y = -2 + 10
2x - y = -8
which is the same equation in standard form.

Graph of  y = 2x + 8 with point (1, 10) on it: